Levi-Civita symbol

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In mathematics, especially in linear algebra, tensor analysis, and differential geometry, the symbol Levi-Civita represents a collection of numbers; defined by the permutation marks of the natural numbers 1, 2,..., n , for some positive integers n . It is named after Italian mathematician and physicist, Tullio Levi-Civita. Other names include permutation symbols , antisymmetric symbols , or alternate symbols , which refer to their antisymmetry properties and definitions in the case of permutations.

The standard letters to show Levi-Civita's symbols are epsilon Greek lowercase ? or ? , or less commonly the Latin letter below e . Index notation allows one to display permutations in a way that is compatible with tensor analysis:

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â{\mathrm{me}}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â1Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â{\mathrm{me}}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â2Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â...Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â{\mathrm{me}}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂnÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; ÂAnnotation\; encoding\; =\; "application/x-tex">\; \{\backslash \; displaystyle\; \backslash \; varepsilon\; \_\; \{i\_\; \{1\}\; i\_\; \{2\}\; \backslash \; dots\; i\_\; \{n\}\}\}\; Â\; Â$

where each the index i ₁ , i ₂ ,..., n takes 1, 2,..., n . There is n ⁿ indexed values ​​ ? _{i 1 i 2} , which can be compiled into n -remaining dimensions. The key that defines a symbol property is total antisymmetry in all indices. When there are two exchanging indices, the same or not, symbols are negotiable:

${\ displaystyle \ varepsilon _ {\ dots i_ {p} \ dots i_ {q} \ point} = - \ varepsilon _ {\ dots i_ {q} \ dots i_ {p} \ dots}.} Â Â$

Jika dua indeks sama, simbolnya nol. Ketika semua indeks tidak sama, kami memiliki:

${\ displaystyle \ varepsilon _ {i_ {1} i_ {2} \ titik-titik i_ {n}} = (- 1) ^ {p} \ varepsilon _ {1 \, 2 \, \ titik-titik n},}  ÂÂ$

where p (called the parity of the permutation) is the number of pairwise pairchange of the index required to decipher i ₁ , i ₂ ,..., i _n into the 1, 2,..., n , and factor (- 1) ^p called a sign or a signature of a permutation. Value ? _{1 2... n} must be specified, otherwise the symbol-specific values ​​for all permutations can not be determined. Most authors choose ? _{1 2... n} = 1 , which means the Levi-Civita symbol is the same as the permutation mark when the index is all not the same. This option is used throughout this article.

The term " n -dimension of the Levi-Civita symbol" refers to the fact that the number of indices in the n symbol corresponds to the dimension of the corresponding vector space, which may be Euclidean or non-Euclidean , for example R ³ or Minkowski space. Levi-Civita symbol values ​​do not depend on tensor and coordinate system metrics. Also, the term specific "symbol" emphasizes that it is not a tensor because of its transformation between coordinate systems; However it can be interpreted as a tensor density.

The Levi-Civita symbol allows the determinant of a square matrix, and crosslinks two vectors in a three-dimensional Euclidean space, which will be expressed in the index notation.

Video Levi-Civita symbol

Definisi

The Levi-Civita symbol is most often used in three and four dimensions, and to some extent in two dimensions, so this is given here before defining a common case.

Two dimensions

Dalam dua dimensi, simbol Levi-Civita didefinisikan oleh:

${\ displaystyle \ varepsilon _ {ij} = {\ begin {cases} 1 & amp; {\ text {if}} (i, j) = (1,2) \ \ -1 & amp; {\ text {if}} (i, j) = (2,1) \\\; \; \, 0 & amp; {\ text {if}} i = j \ end {cases}}}  ÂÂ$

Nilai-nilai dapat diatur ke dalam matriks antisymmetric 2 ± 2:

${\ displaystyle {\ begin {pmatrix} \ varepsilon _ {11} & amp; \ varepsilon _ {12} \\\ varepsilon _ {21} & amp; \ varepsilon _ {22 } \ end {pmatrix}} = {\ begin {pmatrix} 0 & amp; 1 \\ - 1 & amp; 0 \ end {pmatrix}}}  ÂÂ$

The use of two-dimensional symbols is relatively uncommon, although in certain specific topics such as supersymmetry theory and twistor theory appear in the 2-spinner context. Three-dimensional and higher Levi-Civita symbols are used more generally.

Three dimensions

Meaning, ? _ijk is 1 if ( i , j , k ) is a permutation of (1, 2, 3) , -1 if it is a permutation odd, and 0 if any index is repeated. In just three dimensions, the cyclic permutations of (1,2,3) are all permutations, so also the anticyclic permutation is a strange permutation. This means that in 3d enough to take cyclic or anticyclic permutations from (1, 2, 3) and easily obtain all even or odd permutations.

Analog with a 2-dimensional matrix, the 3-dimensional Levi-Civita symbol values ​​can be arranged into the array 3 ÃÆ'â € "3 ÃÆ'â €" 3 :

where i is the depth ( blue : i = 1 ; red : < span> i = 2 ; green : i = 3 ), j is the line and k is the column.

Dalam empat dimensi, simbol Levi-Civita didefinisikan oleh:

${\ displaystyle \ varepsilon _ {ijkl} = {\ begin {cases} 1 & amp; {\ text {if}} (i, j, k, l) {\ text {adalah permutasi bahkan}} (1,2,3,4) \\ - 1 & amp; {\ text {if}} (i, j, k, l) {\ text {adalah permutasi aneh}} ( 1,2,3,4) \\\; \; \, 0 & amp; {\ text {otherwise}} \ end {cases}}}  ÂÂ$

These values ​​can be set into the 4 ÃÆ'â € "4 ÃÆ'â €" 4 array, although in 4 dimensions and higher this is hard to describe.

Thus, this is a sign of permutation in the case of permutations, and zero otherwise.

where the signum function (denoted sgn ) returns its argument mark while dumping the absolute value if it is zero. This formula applies to all index values, and for any n (when n = 0 or n = 1 , this is an empty product). However, computing the above formula naively has the time complexity O ( n ² ) , whereas the sign can be calculated from the permutation parity of its disjoint cycle only in cost O ( n log ( n )) .

Maps Levi-Civita symbol

Properties

A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (covariance rating tensor n ) is sometimes referred to as a permutation tensor .

Under the ordinary transformation rules for tensors, Levi-Civita symbols do not change under pure rotation, consistent with that (by definition) similar across all coordinate systems associated with orthogonal transformations. However, the Levi-Civita symbol is a pseudotensor because under the orthogonal transformation of Jacobian determinant -1, ie. reflection in an odd number of dimensions, it should get a minus sign if it is a tensor. Because it does not change at all, the Levi-Civita symbol, by definition

Source of the article : Wikipedia